Problem 53 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ -\left(r^{2} s t^{3}\right)^{2}\left(s^{4} t\right)^{3} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(-\left(r^{2} s t^{3}\right)^{2}\left(s^{4} t\right)^{3}\) is \( - r^{4} * s^{14} * t^{9} \)

Step 1: Apply the power of a product property

The property of power of a product states that \( (ab)^n = a^n b^n \). So here we apply this principle to both terms: \[ - (r^{2} s t^{3})^{2}(s^{4} t)^{3} = - r^{2*2} * s^{2} * t^{3*2} * s^{4*3} * t^{3} \]

Step 2: Simplify the expression

We then simplify the expression by performing the multiplication in each power: \[ - r^{2*2} * s^{2} * t^{3*2} * s^{4*3} * t^{3} = - r^{4} * s^{2} * t^{6} * s^{12} * t^{3} \]

Step 3: Combine like terms

Now we combine the like terms. The terms \(s^2\) and \(s^{12}\) become \(s^{2+12}\), and similarly, \(t^6\) and \(t^3\) become \(t^{6+3}\): \[ - r^{4} * s^{2} * t^{6} * s^{12} * t^{3} = - r^{4} * s^{14} * t^{9} \]

Key Concepts

Power of a Product

When you have an expression like \((ab)^n\), the power of a product property is your best friend. It lets you distribute the exponent to each factor inside the parentheses. This means \((ab)^n = a^n \cdot b^n\). In our exercise, we apply this rule to the expression \(-\left(r^{2} s t^{3}\right)^{2}\). Here’s how it works:
\[- (r^{2} s t^{3})^{2} = - (r^{2})^2 \cdot (s)^2 \cdot (t^{3})^2 = - r^{4} \cdot s^2 \cdot t^6 \]
Do the same with \((s^{4} t)^{3}\):
\[(s^{4} t)^{3} = (s^{4})^3 \cdot (t)^3 = s^{12} \cdot t^3\]
This step is key to breaking down the expression into smaller, manageable pieces before simplifying it further.

Product of Powers

The product of powers property helps when you combine terms with the same base. If you see something like \(a^m \cdot a^n\), you can simplify it using \(a^{m+n}\). The power of a product informs the breakdown, and now product of powers assists in combining terms. Returning to our expression, we have:

• \(s^2 \cdot s^{12} = s^{2+12} = s^{14}\)
• \(t^6 \cdot t^3 = t^{6+3} = t^9\)

We also retain \(-r^4\) since there are no like terms to combine with r. This method makes merging similar bases straightforward and smoothens our path towards simplification.

Simplifying Expressions

With all the properties applied, your expression should now be easier to manage. Start by combining everything neatly:
\[- r^4 \cdot s^{14} \cdot t^9\]
By efficiently using the power of a product and product of powers properties, you've reduced the original, complex expression to a more elegant form. Remember to keep like bases together, and always handle negative signs carefully. These concepts are fundamental whether you're tackling homework or real-world problems, making expressions simpler and more intuitive to interpret.